Found problems: 788
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)
Consider the set
$$M = \left\{\frac{a}{\overline{ba}}+\frac{b}{\overline{ab}} \, | a,b\in\{1,2,3,4,5,6,7,8,9\} \right\}.$$
a) Show that the set $M$ contains no integer.
b) Find the smallest and the largest element of $M$
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
Find the sum of all distinct four-digit numbers that contain only the digits $1, 2, 3, 4,5$, each at most once.
For a given positive integer $n$, let $f(x) =x^{n}$. Is it possible for the decimal number
$$0.f(1)f(2)f(3)\ldots$$
to be rational? (Example: for $n=2$, we are considering $0.1491625\ldots$)
$P$ and $Q$ are two different non-constant polynomials such that $P(Q(x)) = P(x)Q(x)$ and $P(1) = P(-1) = 2019$. What are the last four digits of $Q(P(-1))$?
Let $S(n)$ be the sum of digits of $n$. Determine all the pairs $(a, b)$ of positive integers, such that the expression
$S(an + b) - S(n)$ has a finite number of values, where $n$ is varying in the positive integers.
Let $a$ be a positive integer such that the number $a^n$ has an odd number of digits in the decimal representation for all $n > 0$. Prove that the number $a$ is an even power of $10$.
The integer $n \geq 1$ does not contain digits: $1,\; 2,\; 9\;$ in its decimal notation. Prove that one of the digits: $1,\; 2,\; 9$ appears at least once in the decimal notation of the number $3n$.
A positive integer is called [i]sabroso [/i]if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$. Find all two-digit sabroso numbers.
Prove that there exist infinitely many positive integers $x,y,z$ for which the sum of the digits in the decimal representation of $~4x^4+y^4-z^2+4xyz$ $~$ is at most $2$.
(Proposed by Gerhard Woeginger, Austria)
Determine all natural numbers $n$ of $4$ digits whose quadruple minus the number formed by the digits of $n$ in reverse order equals $30$.
Let $A$ be a $3$-digit positive integer and $B$ be the positive integer that comes from $A$ be replacing with each other the digits of hundreds with the digit of the units. It is also given that $B$ is a $3$-digit number.
Find numbers $A$ and $B$ if it is known that $A$ divided by $B$ gives quotient $3$ and remainder equal to seven times the sum of it's digits.
Prove that for each prime number $p > 5$ there exists a positive integer n such that $p^n$ ends in $001$ in decimal representation.
A positive integer is a good number, if its base $10$ representation can be split into at least $5$ sections, each section with a non-zero digit, and after interpreting each section as a positive integer (omitting leading zero digits), they can be split into two groups, such that each group can be reordered to form a geometric sequence (if a group has $1$ or $2$ numbers, it is also a geometric sequence), for example $20240327$ is a good number, since after splitting it as $2|02|403|2|7$, $2|02|2$ and $403|7$ form two groups of geometric sequences.
If $a>1$, $m>2$, $p=1+a+a^2+\dots+a^m$ is a prime, prove that $\frac{10^{p-1}-1}{p}$ is a good number.
A two-digit number is [i]nice [/i] if it is both a multiple of the product of its digits and a multiple of the sum of its digits. How many numbers satisfy this property? What is the ratio of the number to the sum of digits for each of the nice numbers?
We define the function $Z(A)$ where we write the digits of $A$ in base $10$ form in reverse. (For example: $Z(521)=125$). Call a number $A$ $good$ if the first and last digits of $A$ are different, none of it's digits are $0$ and the equality: $$Z(A^2)=(Z(A))^2$$ happens. Find all such good numbers greater than $10^6$.\\
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
Given the following list of numbers:
$$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$
where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?
How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?
Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.
Find all natural numbers $n$ for which the sum of digits of $5^n$ equals $2^n$.
Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.
In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$?
$\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7$
A clue “$k$ digits, sum is $n$” gives a number k and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue “Three digits, sum is $23$” has only one answer: $6,8,9$. The clue “Three digits, sum is $8$” has two answers: $1,3,4$ and $1,2,5$.
If the clue “Four digits, sum is $n$” has the largest number of answers for any four-digit clue, then what is the value of $n$? How many answers does this clue have? Explain why no other four-digit clue can have more answers.